Stochastic Stability of ACIMs for Piecewise Expanding $C^{1+\varepsilon}$ Maps
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917401007751168 |
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| author | Rajput, Aparna |
| author_facet | Rajput, Aparna |
| contents | We prove stochastic stability of absolutely continuous invariant measures (ACIMs) for piecewise expanding $C^{1+\varepsilon}$ maps of the interval. For maps $τ$ in the class $\mathcal{T}([0,1]; s, \varepsilon)$, we consider perturbed Frobenius--Perron operators $P_δ= Q_δP_τ$, where $Q_δ$ is a Markov smoothing operator modeling noise of intensity $δ> 0$.
In the generalized bounded variation space $BV_{1,1/p}$, we establish a Lasota--Yorke inequality uniform in $δ$. Consequently, each $P_δ$ admits an invariant density $h_δ\in BV_{1,1/p}$, and $h_δ\to h$ in $L^1$ as $δ\to 0$, where $h$ is the ACIM density of $P_τ$.
Our proof combines the $BV_{1,1/p}$ framework, adapted from recent ACIM existence results, with uniform quasi-compactness and perturbation theory for transfer operators. This establishes stochastic stability under minimal $C^{1+\varepsilon}$ regularity ($\varepsilon > 0$), where the $C^1$ case is known to fail. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03528 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stochastic Stability of ACIMs for Piecewise Expanding $C^{1+\varepsilon}$ Maps Rajput, Aparna Dynamical Systems Primary: 37A05, Secondary: 37E05, 37A10 We prove stochastic stability of absolutely continuous invariant measures (ACIMs) for piecewise expanding $C^{1+\varepsilon}$ maps of the interval. For maps $τ$ in the class $\mathcal{T}([0,1]; s, \varepsilon)$, we consider perturbed Frobenius--Perron operators $P_δ= Q_δP_τ$, where $Q_δ$ is a Markov smoothing operator modeling noise of intensity $δ> 0$. In the generalized bounded variation space $BV_{1,1/p}$, we establish a Lasota--Yorke inequality uniform in $δ$. Consequently, each $P_δ$ admits an invariant density $h_δ\in BV_{1,1/p}$, and $h_δ\to h$ in $L^1$ as $δ\to 0$, where $h$ is the ACIM density of $P_τ$. Our proof combines the $BV_{1,1/p}$ framework, adapted from recent ACIM existence results, with uniform quasi-compactness and perturbation theory for transfer operators. This establishes stochastic stability under minimal $C^{1+\varepsilon}$ regularity ($\varepsilon > 0$), where the $C^1$ case is known to fail. |
| title | Stochastic Stability of ACIMs for Piecewise Expanding $C^{1+\varepsilon}$ Maps |
| topic | Dynamical Systems Primary: 37A05, Secondary: 37E05, 37A10 |
| url | https://arxiv.org/abs/2604.03528 |